Inter won the first leg in Rotterdam with a score of 2-0 after goals from strikers Marcus Thuram and Lautaro Martinez, meaning Robin van Persie and his Feyenoord side had it all to do in the return at San Siro.
Feyenoord started the game eagerly and noted their first shot on target after seven minutes when Aymen Sliti - who Van Persie handed his senior debut - fired a shot at Yann Sommer, but the Swiss stopper had no problem withholding the youngster from a dream debut goal.
A minute later, Inter would play their way through Feyenoord's defense, largely thanks to Marcus Thuram dribbling his way past a lacklustre defensive line and smashing it in the top right corner, giving Inter their first lead after just eight minutes.
The Rotterdam side continued to fight for their spot amongst Europe's elite but could not put the home side under sufficient pressure after efforts from the likes of Wolves loanee Hugo Bueno couldn't beat Sommer.
Fate was on Feyenoord's side after all, however, as Slovakian referee Ivan Kruzliak pointed at the spot after a foul on former Brighton midfielder Jakub Moder. The Polish international took matters into his own hands and calmly slotted the ball in the left-hand corner, handing Feyenoord a lifeline right before the break.
It wouldn't take long for Inter to take back control over the game after the break after Thuram fired a shot at Timon Wellenreuther's goal, but saw his effort fly over the crossbar.
High pressure from Inter resulted in a penalty for the Italians just five minutes after the second-half kick-off after Medhi Taremi tripped over Thomas Beelen's leg relatively easily, but with enough eventual contact for the VAR to let the penalty stand.
Captain Hakan Calhanoglu would take place behind the ball and put Inter back in front in the 51st minute with a well-executed spot kick.
Feyenoord would continue to take the fight to Inter, playing optimistically with the odds heavily against their favour. However, Beelen would soon cause even more trouble after seemingly tripping Marcus Thuram in his own box, were it not for the VAR to intervene and instruct Kruzliak to deny Inter a penalty. Thuram would get a yellow card for simulation after Kruzliak followed the VAR's advice.
The French striker stayed eager and came agonisingly close to his second goal of the night, but he hit the crossbar with a powerful effort from inside the box.
Robin van Persie's men knew a 4-1 deficit was too much to overcome in the closing stages. They simply played out the game, officially ending their most successful Champions League campaign in over twenty years.
Inter now meet Bayern Munich in the quarter-finals of the UEFA Champions League after the German giants beat Bayer Leverkusen 5-0 over two games.